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The Quantile Framework ® for Mathematics & West Virginia A Practical Approach: Linking Assessment to Instruction Presented by: Jason Turner, Manager, Professional Services

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Today ’ s Session What is the Quantile Framework and why do I need to know about it ?

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Today ’ s Session Where can Quantile measures be found?

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Today ’ s Session What can be done with the Quantile Framework ?

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Today ’ s Session How will the Quantile Framework help students?

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Today ’ s Session How can Quantiles be used in the schools and in the classroom?

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The Quantile Framework for Mathematics: Question #1 What is the Quantile Framework for Mathematics? Why do I need to know about it?

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An indicator of mastery of specific skills. A list of mathematical skills. A curriculum for mathematics. A mathematics program. A grade equivalent. Common Misconceptions The Quantile Framework is NOT :

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The Quantile Framework… …is like a “yardstick” ---it’s the way to measure the development of: Students’ mathematical knowledge and Difficulty of math skills and topics

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Developmental scale that locates mathematics skills and concepts within a taxonomy. Characterizes the problem- solver with a measure on the same developmental scale. The Quantile measure uses a common metric for both The Quantile Framework Student Quantile Measure (Q) Mathematical Skill/Topic/Concept Quantile Measure (Q)

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Vocabulary Break: Let’s define an essential concept

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What is a Quantile Measure (QMeasure)? QMeasures are applied to both students and math skills and concepts. Students: The Quantile measure for an individual or student is the level at which he or she is ready for instruction and has knowledge of the prerequisite mathematical concepts and skills necessary to succeed. When applied to students, the QMeasure indicates the student’s readiness for instruction.

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What is a Quantile Measure (QMeasure)? QMeasures are applied to both students and math skills and concepts. Skills & Concepts: When applied to mathematical skills and concepts, QMeasures indicate the task difficulty associated with a skill or concept relative to other mathematical skills and concepts. For example, the skill of rounding whole numbers to a given place value has a QMeasure of 410Q.

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The Quantile Measure… Forecasts the student’s readiness to be successful with the skill after introductory instruction.

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The Quantile Measure… Also refers to the difficulty of a mathematical skill or concept, or how difficult the skill or concept (QTaxon) is compared to others.

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Vocabulary Break: Let’s define another essential concept

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What is a QTaxon? Derived from taxonomy (a category or group) A specific mathematical skill or concept, e.g., to perform long division with remainders is a specific mathematical skill or concept, or QTaxon. Within the Quantile Framework, QTaxon is the formal term for skill or concept.

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Why Do I Need To Know About It? Developmental Scale : Allows districts and schools to track growth. Common Scale : Places task difficulty and student readiness on the same scale. Differentiation & Targeting : Provides math teachers the means to differentiate and target math instruction based on the student’s status.

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The Quantile Framework for Mathematics Where can Quantile measures be found?

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Where can I find the Quantile measure? State End-of-Grade Tests (Texas, West Virginia, North Carolina, & Wyoming) Scholastic Math Inventory (SMI) Voyager V-Math Pearson PASeries

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Where can I find the Lexile measure ? A student’s Lexile measure can be found on the student report for the WESTEST 2 for grades 3 -8 and 10. More information on a student’s Lexile measure based on the WESTEST 2 score is available on the West Virginia Department of Education’s Web site at:

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West Virginia Educational Standards Test 2 (WESTEST 2)

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The Quantile Framework for Mathematics What can be done with the Quantile Framework?

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Two Issues for Math Educators: First, how does a teacher determine student readiness? OR How does a teacher forecast student preparedness to learn mathematical skills or concepts?

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Two Issues for Math Educators: Second, how does a teacher establish the measure of difficulty for various skills in mathematics?

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+ Student Readiness Skill Difficulty Strengthens Instructional Strategies Quantile Scale 700Q 800Q 900Q 1000Q Sally & Jim Raul Jacob, Lyn, Victor

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Success Rates for Students of Different Mathematics Ability with a Skill at a Defined Quantile Measure Student Mathematics Ability Skill - Locate points in all quadrants of the coordinate plane using ordered pairs. Expected Success 350Q 600Q 850Q 1100Q 1350Q 850Q 10% 25% 50% 75% 90%

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Success Rates for Student of Similar Ability with Math Skills of Different Skill Demands Student Mathematics Ability Skill DemandSkill Description Expected Success 750Q250QLocate points on a number line. 90% 750Q500QUse order of operations including parenthesis to simplify numerical expressions 75% 750Q Translate between models or verbal phrases and algebraic expressions. 50% 750Q1000QEstimate and calculate areas with scale drawings and maps. 25% 750Q1250QRecognize and apply definitions and theorems of angles formed when a transversal intersects parallel lines. 10%

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Benefits of the Quantile Framework DETERMINE : student readiness ESTABLISH : task difficulty STRENGTHEN : classroom teaching practices and strategies

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Benefits of the Quantile Framework LINK : student ability to skill difficulty (using a common measure) FORECAST : student success rate (after introductory lessons) TRACK : growth over time

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The Quantile Framework for Mathematics How will the framework help the students and teachers in my district?

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Picturing Mathematics with the Quantile Map

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Gaps in learning: Student Ability 350QSkill Difficulty 850Q What must happen to close this gap? Student: 350QQtaxon: 850Q What must happen to narrow this gap?

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The Quantile Framework: Practical Benefits Helps teachers… Differentiate instruction by linking students with instruction at their level by knowing the Quantile level of the skill or concept and matching it with the Quantile measure of the student Determine student readiness by knowing each student’s Quantile measure Forecast student success with specific skills and concepts

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The Quantile Framework: Practical Benefits Helps teachers… Monitor student math progress by tracking student growth using the Quantile developmental scale Describe pre-requisite skills needed to learn more advanced mathematical concepts Tailor teaching methods in the classroom to ensure a greater level of understanding and application Determine supplementary and upcoming skills for students in need of enrichment

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The Quantile Framework for Mathematics How can Quantiles be used in the classroom?

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Vocabulary Break: Let’s define some more essential concepts

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What is a Prerequisite QTaxon? For any mathematical skill or concept there are prerequisite skills and concepts necessary to succeed. When we refer to Prerequisite QTaxons, we mean those skills that are necessary to be successful with the skill in question. For example, to successfully perform division with remainders, one must be able to successfully perform simple division without remainders.

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What’s an Impending QTaxon? Every mathematical skill or concept is building toward another skill or concept. For example, the multiplication of single digit numbers comes after the addition of double digit numbers, and is, therefore, an impending skill… Impending QTaxons are the skills that ‘come next’ or are the next steps in the Quantile taxonomy

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What’s a Supplemental QTaxon? A Supplemental QTaxon is a mathematical skill or concept which broadens, enhances or deepens knowledge of a skill or concept, but is not necessary for success with that skill or concept. For example, a Supplemental QTaxon of “Calculate the areas of triangles, parallelograms, trapezoids, circles and composite figures”, is “Use models to develop formulas for finding areas of triangles, parallelograms, trapezoids, and circles” This skill enhances understanding of the QTaxon in question, but is not necessary for success.

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An Essential Feature of the Quantile Framework: The Knowledge Cluster! Precursory skills to ensure prior knowledge Supporting skills to connect and add depth across content strands Quantile measures to address the degree of difficulty for each skill

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The Power of the Knowledge Cluster To inform instruction and address the needs and abilities of each student To add depth and breadth to concepts in mathematics To adjust Individual Education Plans according to Quantile measures

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The Knowledge Cluster Accessing the knowledge cluster

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Knowledge Clusters

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Q Taxon ID numbers

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A QTaxon and It’s Knowledge Cluster QT-M-254 Investigate and determine the relationship between the diameter and the circumference of a circle and the value of pi; calculate the circumference of a circle.

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Subtract 2- and 3-digit numbers with regrouping.

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Quantile Resources

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Math Skill Database QTaxon Search Knowledge Clusters Curriculum Alignment Textbook lesson alignment Instructional Resources “Leading/Reverse” QTaxon Search Math At Home

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Quantile Professional Development helps put the Quantile Framework to work for you and your students. Our Master Facilitators empower you with the knowledge and skills to implement the Quantile Framework in your classroom. QUANTILES IN YOUR DISTRICT:Understand how the Quantile Framework for Mathematics was developed and learn to use Quantiles to understand task difficulty. STEP 1: UNDERSTANDING AND USING THE QUANTILE FRAMEWORK FOR MATHEMATICS:Hands on practice using the tools for differentiating mathematics to meet standards. Learn to use Quantiles to understand task difficulty and student readiness. STEP 2: PUTTING QUANTILES TO WORK IN YOUR CLASSROOM: This workshop offers strand specific, in-depth study of the connections along the mathematics curriculum. For more information, call LEXILES/ or Quantile Professional Development

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Contact Information Jason Turner Manager, Professional Development MetaMetrics Inc.

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